Different from a canonical DP mixture model, the observed data \(\mathbf {z}_{0:T}\) , \(\mathbf {u}_{0:T}\) , and \(\mathbf {o}_{0:T}\)  are not independent samples given variables \(\mathbf {X}_{0:T}\) , \(\mathbf {L}\) , and \(\pi \) , but are correlated through the factor graph.
Therefore, the inference involves computing maximum likelihood over factor graphs.
When both associations and variables are to be established, standard approaches alternate between assigning data and optimizing variables.
In the case of known object number \(M\) , K-means has a deterministic data association, while expectation-maximization associates data in a probabilistic way [1]}.
When the number of objects is not known a priori and DP is used as prior, Markov Chain Monte Carlo methods (e.g. Gibbs sampling) or variational inference algorithms are widely used [1]}.
However, in these algorithms, the likelihood of each label \(y_t^k\)  to be any underlying object \(\mathbf {L}\)  needs to be computed and tracked all the time. The algorithm will need to go through all of the data multiple times to converge to a steady state distribution.
The large scale and strong dependence of data in our problem make such approaches inappropriate.
